Yesterday, in a feat of mathematical endurance, I attended all four talks in the Current Events Bulletin. The Current Events Bulletin, started in 2003, is a neat idea for the joint meetings: prominent researchers give expository talks about important areas of research (generally not including their own research) for a general mathematical audience. This year, Carina Curto talked about topology and the neural code, Yuval Peres talked about the dynamics of abelian sandpiles, Timothy Gowers talked about probabilistic combinatorics and recent results in combinatorial design theory, and Amie Wilkinson talked about Lyapunov exponents. All four also contributed short overview papers to a booklet available to the audience. (The booklet is also available online as a pdf.) Interestingly, each talk was about 45 minutes long with a short break halfway through to allow people to escape if necessary without causing too much commotion.
These summaries are based only on my notes from the talks. I apologize for any errors or misinterpretations.
I probably got the most out of Carina Curto’s talk both because it was the first talk, so I was not yet fatigued, and because I went in knowing almost nothing about the topic. Topology comes into neuroscience in several guises, including topological data analysis, network theory, and understanding the neural code, the way the brain encodes information. Curto focused on the last topic, particularly as it relates to information about place. Neurons called place cells fire in response to an animal’s location in its environment. Even more interesting to me are grid cells. These cells fire based on the animal’s location in the environment, but the same neuron fires in response to several locations. The locations that cause one neuron to fire form a hexagonal lattice in the environment. Topology comes into studying both place and grid cells. For place cells, the places that stimulate each neuron form an open cover of the environment. (At the phrase “open cover,” topologists instinctively prick up their ears.) By looking at the firing of the place cells, the topology of the environment can be reconstructed based only on that information. For grid cells, there is a “fundamental domain” of the environment; the grid cells respond as if the environment is a torus. Weird!
Yuval Peres spoke about sandpiles. I’ve read a little about them before, including a great Nautilus article by Jordan Ellenberg. The idea of an abeliean sandpile is that you start with a number of grains of sand on one square in a grid. When the square has four grains of sand on it, it topples over, giving one grain of sand to each neighboring grid cell. (It’s abelian because, although it is not obvious, when two squares both have 4 or more grains of sand, the order of toppling doesn’t matter.) Peres talked about this problem and variations thereof. The starting grid can be seeded instead of empty. The “sand” can be made continuous instead of discrete, so a square can keep an integer amount of sand and equally distribute a fractional amount to each neighbor (in his article about it, he appealingly described this model as maple syrup in the grid of a waffle). Then there was “rotor routing,” which encodes a “nonrandom random walk” on the grid by labeling the squares with directions as the walker walks through them. Peres included a tantalizing open question at the end of the talk. He showed us a two-dimensional sandpile and a slice of a three-dimensional one. Outside of a neighborhood of the origin, the pictures were the same. Not similar, but pixel-by-pixel identical. There is currently no proof of why that would be.
Timothy Gowers spoke about Peter Keevash’s recent work on combinatorial design theory, especially ideas of probabilistic combinatorics. The standard example question is the Kirkman schoolgirl problem. There are 15 schoolgirls, and they walk to class in five rows of three. Is there a way they can walk so that over the course of the week, no two of them walk abreast twice? The general problem is to determine whether, in a set of n people, there is any way to form groups of people so that every set of s people is contained in exactly one set of r people? Gowers introduced some naive ways to approach the topic and explained why they didn’t quite work but how some of them could be tweaked to
Amie Wilkinson’s topic was the one I was most familiar with already, although because I am familiar due to more to my spouse’s rather than my own research (I am a dynamicist-in-law), my knowledge has some gaps. Her talk title is a question I have wondered about several times: What are Lyapunov exponents, and why are they interesting? The example she started with was the successive barycentric division of an equilateral triangle. As you find the barycenters of the triangles, is there some kind of limiting triangle shape? To make this question more precise, when you take a random walk through the triangle, choosing one of the six subtriangles at each step, how does the “aspect ratio” change? In this case, it goes to zero, so the triangles are getting very thin and needlelike. The rate at which they get needle-like is the Lyapunov exponent. After explaining what they were, Wilkinson gave a general overview of some of the places they pop up in Fields medalist Artur Avila’s work: ergodic theory, translation surfaces, and 1-dimensional Schrödinger operators.
On a practical note, the talks were in a dim, warm room, and by the end of the second one, I was sore from sitting so long. But I took a deep breath and powered through. In other words, this was a professional on a closed course. Do not attempt at home.